Step by step
The question
Question: find the exact value of tan 30°.
1 Draw the unit circle
The unit circle has its center at 0 and radius 1.
We turn 30° from the positive x-axis, against the clock.
The ray meets the circle at the point P.
Here is the key fact.
P is the point (cos 30°, sin 30°).
Across is cos.
Up is sin.
2 Use the special triangle
Draw a line from P straight down to the x-axis.
It makes a right triangle.
Its longest side goes from the center to P.
It is a radius: 1.
The angles are 30°, 60° and 90°.
The short side x is across from the 30° angle.
From the special triangles lesson: the sides are x, x root 3 and 2x.
The longest side, 2x, is 1.
So 2x = 1.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
So x = 1/2: the short side.
The other side is x root 3: 1/2 times √3.
That is √3/2.
3 Read the point
Across, from the center to under P: √3/2.
Up, from the x-axis to P: 1/2.
So P = (√3/2, 1/2).
Careful!
Tan is sin over cos: the up number divided by the across number.
4 Divide
Tan is sin over cos: up divided by across.
Now put the numbers in: sin = 1/2 and cos = √3/2.
To divide by a fraction, we multiply by its flip.
A 2 at the bottom and a 2 on top: 2/2 is 1.
Cross them out.
What is left: 1 over √3.
Books write it with no root at the bottom: times √3 on top and bottom.
√3 times √3 is 3.
So tan 30° = √3/3.
5 Check
Let's check: tan times cos must give sin back.
√3/3 times √3/2.
We get 1/2: that is sin 30°.
So it is right!
Let's look back.
First: draw the unit circle.
Then: use the special triangle.
Then: read the point.
Then: divide.
Last: check.